1 fmr) 
1 [(mr) 
452 
Proceedings of the Royal Society 
f r - ft - 1 + 2(8, + -1159315)] + - 1 + 2(S 2 + -1159315)] + 
-log 
L( 
mr , m 3 r 3 m 5 r 5 0 
+ -^r-r + ^ ^ + &C 
mr\ 2 2 2 .4 2 2 .4 2 .6 
■) 
2 y»2 
=13 - o¥ - 3 + 2 ( S i + -1159315)] + - 6(S 2 - -1159315)] + &c. 
m“r 
2 r 2 2 2 
2 2 .4 2L 
, 1/1 3m 2 r 2 , 5 m 4 r 4 « \ 
~ l0S mA 2" + “PT PTPT6 + &C 'J 
For an illustration of Case II. with. i= 1, suppose ma to be very 
small. Remarking that S x = 1, we have 
■vr _ - wal/ (ma) 
l]M ” 
1 m 2 a 2 
log— - A + -1159 
~2~ 
L mu 
1 m 2 a 2 
log i_ + i + -1159 
~2 
L mu 
= l+m 2 s 2 6ogi- + -1159 N ) . (12); 
\ ma / 
Hence in this case at all events N >i 2 ; and the angular velocity of 
the slow wave, in the reverse direction to that of the liquid’s revo- 
lution, is 
-» = io.m 2 a 2 (log— +-1159 N ] . . (43). 
V mu ) 
This is very small in comparison with 
2o> + UmW f log A + -11 59^ . . (44), 
V ma / 
the angular velocity of the direct wave ; and therefore clearly if the 
initial normal velocity of the surface when left free after being dis- 
placed from its cylindrical figure of equilibrium be zero or anything 
small, the amplitude of the quicker direct wave will be very small 
in proportion to that of the reverse slow one. 
Case III. 
A slightly disturbed vortex column in liquid extending through 
all space between two parallel planes ; the undisturbed column con- 
sisting of a core of uniform vorticity (that is to say, rotating like a 
solid) surrounded by irrotationally revolving liquid with no slip at 
